Martin, I thank you for complimenting my civility and friendliness,
but I must admit you disappointed me.
I know we disagree, but from you I expected a more serious response.
Let me reply to four of your points, if I may.
1) The timeless truth of Lagrange's theorem is demonstrated by the
possibility of arranging piles of pebbles into 1, 2, 3, or 4 squares,
even before there were people.
2) What I am saying about the social nature of mathematical existence
is obvious.
3) What I am saying doesn't help in work on Godel's Legacy.
4) I was irresponsible to omit Godel's Legacy from my book.
I am trying to paraphrase accurately, I hope it isn't a crucial
issue if I have failed to quote you exactly.
1) Another member of the list has already commented on this.
As I remember, Lagrange's theorem is a statement about an arbitrary
positive integer n, or if you prefer, about all positive integers n. I don't
think you will find an arbitrary positive integer n or all positive
integers n in any pile of pebbles. That is why you can't prove
Lagrange's theorem with pebbles. Where do you find an arbitrary
positive integer? It's an idea, a concept, that is understood by many
people. Proving Lagrange's theorem, or even stating it or understanding
it, is an intersubjective or social activity of such people. Yes,
it has applications and consequences for piles of pebbles. No,
it is not contained in piles of pebbles. Calling this simple statement
obvious, or postmodern, or relativist, suggests that the person
doing such name-calling doesn't wish to respond to what I am saying.
You don't stoop to such childishness, but some others do.
2) You misquote me as saying mathematics is a social activity,
and then you call your own misquotation "obvious." I agree. It's
obvious. It would not be worth writing or speaking if one's thoughts
were so trivial. I am saying that the existence, the reality (and they
are real) of mathematical objects is neither physical, subjective
or trranscendental, but in the shared thinking and communication of humn
beings. Disagree, please, and please argue and show why you're sure this
isn't so. It obviously isn't obvious, because so many
fom'ers just don't get it. Calling it relativist, postmodern,
obvious, etc. only suggests a disinclination to actually respond
to what I am actually saying. Ditto to calling me a comedian,
a terrorist, or saying one is shocked at the publicizing of
my book which is "shot through with elementary blunders". (The
blunder he discovered turned out to be no blunder.)
3) It doesn't help us work on Godel's Legacy.
True.
Could there be interesting statements about the nature of
mathematics that don't help on Godel's Legacy?
I think so.
For instance, "The natural numbers arose from the practise
of counting, and later from the needs of the market place."
You may agree or disagree. I think this is a statement of
some interest, worthy of not being immediately dumped into
the dumpster. Yet it doesn't, so far as I can see, help
with Godel's Legacy.
I could make up more examples, but you get the idea.
Your statement was in response to my comment, that it
is remarkable that I am simultaneously called obvious
and crazy. You answer seems to suggest that anything
that doesn't contribute to Godel's Legacy is garbage.
You can't really believe that.
Of course, it would make sense to ask, if my statements don't contribute to
Godel's Legacy, do they contribute to anything?
What value can they possibly have, in view of
that non-contribution? My belief is that the question
of the sense in which mathematical objects exist is
an interestig one. Not to everyone. Not to you, as you state plainly.
To you it's a non-problem, as you put it.
You even say it can be left to the philosophers.
Fine. No reason you should be interested in that.
But does that mean nobody may be interested in it?
You can say, if you like, that the question of
the sense in which mathematical objects exist is inadmissible
on the fom list. Maybe fom'ers don't want to hear about
that kind of thing. But why the name-calling?
4) Was I irresponsible to omit Godel's Legacy from my book?
First of all, let me tell you plainly that if I had known
it was important to you to include it, I would have gladly
done so. We had a conversation at CUNY, but if you told me
then that I had to put Godel's Legacy in my book, somehow I didn't
grasp your message. Maybe you actually never told me that.
In the absence of any specific requests from you or other
devotees of Godel's Legacy, I had to use my own brain to
write the book. Now since you were so frank to tell me
that my problem is of no interest to you, perhaps I can dare
to be equally frank, and say that Godel's Legacy does not
grab me by the throat. In other words, I find it interesting,
but only mildly interesting. And I don't see how this is
the foundations of mathematics.
It looks to me like an interesting, intractable problem in
set theory. Of course, every branch of mathematics has its
interesting, intractable problems, few or none of which I
found it necessary or helpful to put into my book.
I know that fom is not a branch of mathematics. Is it a
branch of anything? Or just that rarity, a branch of
learning sui generis (the only one of its kind.)
Anyhow, I think that putting Godel's Legacy in or leaving it
out was a matter for my best authorial judgemnt. I'm sure
specialists in many branches of mathematics may feel I
erred to leave them out too. I suspect they wouldn't feel
that I committed a crime by leaving out Godel's Legacy!
But you may say, I was writing on the philosophy of mathematics,
and Godel's Legacy is philosophy of mathematics. Well, to
me meaning, existence, truth, and such matters are of philosophical
interest. Godel's Legacy, as far as I understand it, is a
research project in set theory. As such, in my opinion, putting
it in or leaving it out was optional, up to me, not irresponsible.
It's unreasonable for you to declare you have no interest in
the sense in which mathematical objects exist, and in the
same letter call me irresponsible because I don't see the
necessity of putting Godel's Legacy in my book.
One more thing. To my claim that I used my philosophical viewpoint
to deal with a number of puzzles about mathematics, you replied with
one word. Was it "unsuccessfully"? Or the equivalent.
I have to admit that hurt my feelings. I am going to ask you ro
reread the section on inventing and discovering. It's only a few
pages, it won't take you 15 minutes. This is
an old controversial topic between Platonists and Formalists, which never
gets anywhere. I used the actual language of mathematicians in
talking about their own and each other's work, to show that
mathematical progress is a closely interwoven combination of
inventing and discovering, giving a clear explanation of each term.
Do you think that was obvious? The mathematicians to whom I have
told it in dozens of talks all over the place, without exception
found it new, interesting, thought-provoking. Of course, it
could still be obvious to you. And certainly it doesn't
do anything for Godel's Legacy. Does that really mean it's
just worthless, zero, garbage? I would really appreciate an
answer to this, even if you don't want to bother with the rest of
this letter.
Best regards as always,
Reuben Hersh